Low-twist chiral optical layers and related fabrication methods

ABSTRACT

An optical element includes a first and second stacked birefringent layers. The first birefringent layer includes local anisotropy patterns having respective relative orientations that vary over a first thickness between opposing faces of the first birefringent layer to define a first twist angle. The second birefringent layer includes local anisotropy patterns having respective relative orientations that vary over a second thickness between opposing faces of the second birefringent layer to define a second twist angle different than the first twist angle. Related devices and fabrication methods are also discussed.

CLAIM OF PRIORITY

The present application is a continuation of U.S. patent applicationSer. No. 12/596,189, filed Apr. 27, 2010, which is a 35 U.S.C. §371national phase application of PCT International Application No.PCT/US2008/004888, entitled “Low-Twist Chiral Liquid CrystalPolarization Gratings and Related Fabrication Methods”, having aninternational filing date of Apr. 16, 2008, which claims priority toU.S. Provisional Patent Application No. 60/912,044, entitled “Low-TwistChiral Liquid Crystal Polarization Gratings and Related FabricationMethods”, filed Apr. 16, 2007, the disclosures of which are herebyincorporated herein by reference as set forth in their entireties.

FIELD OF THE INVENTION

The present invention relates to optical layers and related methods offabrication.

BACKGROUND OF THE INVENTION

Liquid crystals may include liquids in which an ordered arrangement ofmolecules exists. Typically, liquid crystal (LC) molecules may beanisotropic, having either an elongated (rod-like) or flat (disk-like)shape. As a consequence of the ordering of the anisotropic molecules, abulk LC often exhibits anisotropy in its physical properties, such asanisotropy in its mechanical, electrical, magnetic, and/or opticalproperties.

As a result of the rod-like or disk-like nature, the distribution of theorientation of LC molecules may play an important role in opticalapplications, such as in liquid crystal displays (LCDs). In theseapplications, LC alignment may be dictated by an alignment surface. Thealignment surface may be treated so that the LC aligns relative to thesurface in a predictable and controllable way. In many cases, thealignment surface may ensure a single domain through the LC device. Inthe absence of a treated alignment surface, the LC may have many domainsand/or many discontinuities in orientation. In optical applications,these domains and discontinuities may cause scattering of light, leadingto a degradation in the performance of the display.

Polarization gratings may be used to periodically affect the localpolarization state of light traveling therethrough (as opposed toaffecting the phase or amplitude as in conventional gratings). Forexample, switchable liquid crystal polarization gratings (LCPGs) can beused to implement an intensity modulator that can operate on unpolarizedlight. More particularly, such switchable LCPGs may be used to achieverelatively high contrast modulation of unpolarized light with arelatively narrow bandwidth (such as a laser), for example, inapplications including projection displays and light-shutters. Forinstance, some conventional LCPGs may modulate light with a contrastratio of greater than about 200:1 in the 0^(th)-order for light having abandwidth of <5%. However, the contrast modulation of conventional LCPGsmay degrade when applied to modulate broadband light (such as fromLEDs), which may be important in many applications.

SUMMARY OF THE INVENTION

According to some embodiments of the present invention, a polarizationgrating includes a substrate and a first polarization grating layer onthe substrate. The first polarization grating layer includes a molecularstructure that is twisted according to a first twist sense over a firstthickness defined between opposing faces of the first polarizationgrating layer.

In some embodiments, respective relative orientations of molecules ofthe first polarization grating layer may be rotated by a first twistangle over the first thickness such that a local anisotropy pattern ofthe first polarization grating layer may have a continuously variablephase shift over the first thickness. Also, the substrate may be areflective substrate.

In other embodiments, the polarization grating may further include asecond polarization grating layer on the first polarization gratinglayer. The second polarization grating layer may include a molecularstructure that is twisted according to a second twist sense opposite thefirst twist sense over a second thickness defined between opposing facesof the second polarization grating layer. In particular, respectiverelative orientations of molecules of the first polarization gratinglayer may be rotated by a first twist angle over the first thickness,and respective relative orientations of molecules of the secondpolarization grating layer may be rotated by a second twist angle overthe second thickness.

In some embodiments, the second twist angle may be an opposite anglethan the first twist angle. As such, a local anisotropy pattern of thesecond polarization grating layer may have a continuously variable phaseshift over the second thickness that may be opposite to that of a localanisotropy pattern of the first polarization grating layer over thefirst thickness. For example, the second twist angle may be about +70degrees, while the first twist angle may be about −70 degrees.

In other embodiments, the respective orientations of the molecules ofthe first and second polarization grating layers may be aligned along aninterface therebetween. The substrate may be a transmissive substrate.

In some embodiments, the first polarization grating layer may be a firstchiral liquid crystal layer including chiral liquid crystal moleculestherein having the first twist sense. The second polarization gratinglayer may be a second chiral liquid crystal layer including chiralliquid crystal molecules therein having the second twist sense.

In other embodiments, at least one of the first and second polarizationgrating layers may be a polymerizable liquid crystal layer.

In some embodiments, another of the first and second polarizationgrating layers may be a non-reactive liquid crystal layer. For example,the non-reactive liquid crystal layer may be a nematic liquid crystallayer.

In other embodiments, the first and/or second thicknesses of the firstand second polarization grating layers may be configured to providehalf-wave retardation of light within an operational wavelength range ofthe polarization grating.

In some embodiments, the polarization grating may further include afirst alignment layer on the substrate having a first periodic alignmentcondition therein. The first polarization grating layer may be on thefirst alignment layer, and molecules of the first polarization gratinglayer may be aligned according to the first periodic alignment conditionof the alignment layer.

In other embodiments, the polarization grating may include a secondalignment layer having a second periodic alignment condition therein onthe first polarization grating layer opposite the first alignment layer.The first polarization grating layer may be a non-reactive liquidcrystal layer between the first and second alignment layers. Thenon-reactive liquid crystal layer may include liquid crystal moleculeshaving respective relative orientations that are rotated over thethickness by a twist angle that is different from a relative phase anglebetween the first and second periodic alignment conditions of the firstand second alignment layers.

According to other embodiments of the present invention, a method offorming a polarization grating includes forming a substrate, and forminga first polarization grating layer on the substrate. The firstpolarization grating layer includes a molecular structure that istwisted according to a first twist sense over a first thickness definedbetween opposing faces of the first polarization grating layer.

In some embodiments, the first polarization grating layer may be formedto include molecules having respective relative orientations that arerotated by a first twist angle over the first thickness such that alocal anisotropy pattern of the first polarization grating layer has acontinuously variable phase shift over the first thickness. Also, thesubstrate may be a reflective substrate.

In other embodiments, a second polarization grating layer may be formedon the first polarization grating layer. The second polarization gratinglayer may include a molecular structure that is twisted according to asecond twist sense opposite the first twist sense over a secondthickness defined between opposing faces of the second polarizationgrating layer. In particular, the first polarization grating layer maybe formed such that respective orientations of molecules of the firstpolarization grating layer may be rotated by a first twist angle overthe first thickness. Likewise, the second polarization grating layer maybe formed such that respective orientations of molecules of the secondpolarization grating layer may be rotated by a second twist angle overthe second thickness.

In some embodiments, the second polarization grating layer may be formedon the first polarization grating layer such that the respectiveorientations of the molecules of the first and second polarizationgrating layers are aligned along an interface therebetween. Thesubstrate may be a transmissive substrate.

In other embodiments, forming the first polarization grating layer mayinclude doping a first liquid crystal layer with chiral liquid crystalmolecules having the first twist sense. Also, forming the secondpolarization grating layer may include doping a second liquid crystallayer with chiral liquid crystal molecules having the second twistsense.

In some embodiments, a first alignment layer may be formed on thesubstrate. The first alignment layer may have a first periodic alignmentcondition therein. The first polarization grating layer may be formeddirectly on the first alignment layer such that molecules of the firstpolarization grating layer are aligned according to the first periodicalignment condition. Then, the second polarization grating layer may beformed on the first polarization grating layer.

In other embodiments, the first polarization grating layer may be apolymerizable liquid crystal layer. The polymerizable liquid crystallayer may be photo-polymerized on the first alignment layer prior toforming the second polarization grating layer thereon.

In some embodiments, a second alignment layer may be formed on a secondsubstrate. The second alignment layer may have a second periodicalignment condition therein. The second substrate including the secondalignment layer thereon may be assembled adjacent to the firstpolarization grating layer to define a gap between the second alignmentlayer and the first alignment layer, and the second polarization gratinglayer may be formed in the gap. For example, the second polarizationgrating layer may be a non-reactive liquid crystal layer.

In other embodiments, the second polarization grating layer may be apolymerizable liquid crystal layer.

In some embodiments, the first and/or second thicknesses of the firstand second polarization grating layers may be configured to providehalf-wave retardation of light within an operational wavelength range ofthe polarization grating.

According to further embodiments of the present invention, a switchablepolarization grating includes a first substrate including a firstperiodic alignment condition, a second substrate including a secondperiodic alignment condition, and a liquid crystal layer between thefirst and second substrates. The liquid crystal layer includes liquidcrystal molecules having respective relative orientations that arerotated over a thickness defined between opposing faces thereof by atwist angle that is different from a relative phase angle between thefirst and second periodic alignment conditions.

In some embodiments, the second periodic alignment condition may be outof phase relative to the first periodic alignment condition.

In other embodiments, the liquid crystal layer may be a nematic liquidcrystal layer including a chiral dopant therein having a twist senseconfigured to twist a molecular structure of the liquid crystal layer bythe twist angle over the thickness thereof

In some embodiments, the liquid crystal molecules may be alignedaccording to the first and second alignment conditions of the first andsecond alignment layers at respective interfaces therebetween. Theliquid crystal molecules may also be rotated by the twist angle over thethickness of the liquid crystal layer such that the liquid crystal layercomprises an elastic-energy strain therein.

In other embodiments, the phase angle may be about 70° to about 360°.Also, the twist angle may be about 70° to about 360°.

According to still further embodiments of the present invention, amethod of fabricating a switchable polarization grating includes forminga first substrate including a first periodic alignment condition,forming a second substrate including a second periodic alignmentcondition, and forming a liquid crystal layer on the first and secondsubstrates. The liquid crystal layer includes liquid crystal moleculeshaving respective relative orientations that are rotated over athickness defined between opposing faces thereof by a twist angle thatis different from a relative phase angle between the first and secondperiodic alignment conditions.

In some embodiments, a first alignment layer may be formed on the firstsubstrate and patterned to define the first periodic alignment conditiontherein. Also, a second alignment layer may be formed on the secondsubstrate and patterned to define the second periodic alignmentcondition therein out of phase relative to the first periodic alignmentcondition.

In other embodiments, the liquid crystal layer may be a nematic liquidcrystal layer. The nematic liquid crystal layer may be doped with achiral molecule having a twist sense configured to twist a molecularstructure of the liquid crystal layer by the twist angle over thethickness thereof.

In some embodiments, the liquid crystal layer may be formed such thatthe molecules thereof may be aligned according to the first and secondalignment conditions of the first and second alignment layers atrespective interfaces therebetween and may be rotated by the twist angleover the thickness of the liquid crystal layer such that the liquidcrystal layer has an elastic-energy strain therein.

Other devices and/or methods of fabrication according to someembodiments will become apparent to one with skill in the art uponreview of the following drawings and detailed description. It isintended that all such additional methods and/or devices be includedwithin this description, be within the scope of the invention, and beprotected by the accompanying claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a perspective view illustrating polarization gratingsaccording to some embodiments of the present invention,

FIG. 1B is a top view illustrating polarization gratings according tosome embodiments of the present invention.

FIG. 1C is a side view illustrating polarization gratings according tosome embodiments of the present invention.

FIG. 1D is a side view illustrating polarization gratings according tofurther embodiments of the present invention.

FIGS. 2 is a diagram illustrating a model used to simulate properties ofpolarization gratings according to some embodiments of the presentinvention.

FIGS. 3, 4A and 4B are graphs illustrating properties of polarizationgratings according to some embodiments of the present invention based onsimulation results.

FIGS. 5A-5E are cross-sectional views illustrating methods offabricating polarization gratings and devices so fabricated according tosome embodiments of the present invention.

FIGS. 6A and 6B are graphs illustrating properties of polarizationgratings according to some embodiments based on experimental results.

FIGS. 7A-7D are cross-sectional views illustrating methods offabricating polarization gratings and devices so fabricated according tofurther embodiments of the present invention.

FIGS. 8A-8E are graphs illustrating electro-optical characteristics ofpolarization gratings according to further embodiments of the presentinvention.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

The present invention is described more fully hereinafter with referenceto the accompanying drawings, in which embodiments of the invention areshown. This invention may, however, be embodied in many different formsand should not be construed as limited to the embodiments set forthherein. Rather, these embodiments are provided so that this disclosurewill be thorough and complete, and will fully convey the scope of theinvention to those skilled in the art. In the drawings, the size andrelative sizes of layers and regions may be exaggerated for clarity.Like numbers refer to like elements throughout.

It will be understood that, although the terms first, second, third etc.may be used herein to describe various elements, components, regions,layers and/or sections, these elements, components, regions, layersand/or sections should not be limited by these terms. These terms areonly used to distinguish one element, component, region, layer orsection from another region, layer or section. Thus, a first element,component, region, layer or section discussed below could be termed asecond element, component, region, layer or section without departingfrom the teachings of the present invention.

Spatially relative terms, such as “beneath”, “below”, “lower”, “under”,“above”, “upper” and the like, may be used herein for ease ofdescription to describe one element or feature's relationship to anotherelement(s) or feature(s) as illustrated in the figures. It will beunderstood that the spatially relative terms are intended to encompassdifferent orientations of the device in use or operation in addition tothe orientation depicted in the figures. For example, if the device inthe figures is turned over, elements described as “below” or “beneath”or “under” other elements or features would then be oriented “above” theother elements or features. Thus, the exemplary terms “below” and“under” can encompass both an orientation of above and below. The devicemay be otherwise oriented (rotated 90 degrees or at other orientations)and the spatially relative descriptors used herein interpretedaccordingly. In addition, it will also be understood that when a layeris referred to as being “between” two layers, it can be the only layerbetween the two layers, or one or more intervening layers may also bepresent.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the invention. Asused herein, the singular forms “a”, “an” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. It will be further understood that the terms “comprises”and/or “comprising,” when used in this specification, specify thepresence of stated features, integers, steps, operations, elements,and/or components, but do not preclude the presence or addition of oneor more other features, integers, steps, operations, elements,components, and/or groups thereof. As used herein, the term “and/or”includes any and all combinations of one or more of the associatedlisted items.

It will be understood that when an element or layer is referred to asbeing “on”, “connected to”, “coupled to”, or “adjacent to” anotherelement or layer, it can be directly on, connected, coupled, or adjacentto the other element or layer, or intervening elements or layers may bepresent. In contrast, when an element is referred to as being “directlyon,” “directly connected to”, “directly coupled to”, or “immediatelyadjacent to” another element or layer, there are no intervening elementsor layers present.

Embodiments of the invention are described herein with reference tocross-section illustrations that are schematic illustrations ofidealized embodiments (and intermediate structures) of the invention. Assuch, variations from the shapes of the illustrations as a result, forexample, of manufacturing techniques and/or tolerances, are to beexpected. Thus, embodiments of the invention should not be construed aslimited to the particular shapes of regions illustrated herein but areto include deviations in shapes that result, for example, frommanufacturing. Accordingly, the regions illustrated in the figures areschematic in nature and their shapes are not intended to illustrate theactual shape of a region of a device and are not intended to limit thescope of the invention.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which this invention belongs. It will befurther understood that terms, such as those defined in commonly useddictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art and/orthe present specification and will not be interpreted in an idealized oroverly formal sense unless expressly so defined herein.

It will be understood by those having skill in the art that, as usedherein, a “transmissive” or “transparent” substrate may allow at leastsome of the incident light to pass therethrough. Accordingly, thetransparent substrate may be a glass substrate in some embodiments. Incontrast, a “reflective” substrate as described herein may reflect atleast some of the incident light. Also, “polymerizable liquid crystals”may refer to relatively low-molecular weight liquid crystal materialsthat can be polymerized, and may also be described herein as “reactivemesogens”. In contrast, “non-reactive liquid crystals” may refer torelatively low-molecular weight liquid crystal materials that may not bepolymerized.

Embodiments of the present invention are described herein with referenceto liquid crystal (LC) materials and polarization gratings composedthereof. As used herein, the liquid crystals can have a nematic phase, achiral nematic phase, a smectic phase, a ferroelectric phase, and/oranother phase. In addition, a number of photopolymerizable polymers maybe used as alignment layers to create the polarization gratingsdescribed herein. In addition to being photopolymerizable, thesematerials may be inert with respect to the LC, should provide stablealignment over a range of operating temperatures of the LC device (e.g.,from about −50° C. to about 100° C.), and should be compatible withmanufacturing methods described herein. Some examples ofphotopolymerizable polymers include polyimides (e.g., AL 1254commercially available from JSR Micro, Inc (Sunnyvale, Calif.)), NissanRN-1199 available from Brewer Science, Inc. (Rolla, Mo.), and cinnamates(e.g., polyvinyl 4-methoxy-cinnamate as described by M. Schadt et al.,in “Surface-Induced Parallel Alignment of Liquid Crystals by LinearlyPolymerized Photopolymers,” Jpn. J. Appl. Phys., Vol. 31 (1992), pp.2155-2164). Another example of a photopolymerizable polymer isStaralign™, commercially available from Vantico Inc. (Los Angeles,Calif.). Further examples include chalcone-epoxy materials, such asthose disclosed by Dong Hoon Choi and co-workers in “Photo-alignment ofLow-molecular Mass Nematic Liquid Crystals on PhotochemicallyBifunctional Chalcone-epoxy Film by Irradiation of a Linearly PolarizedUV,” Bull. Korean Chem. Soc., Vol. 23, No. 4 587 (2002), and coumarinside chain polyimides, such as those disclosed by M. Ree and co-workersin “Alignment behavior of liquid-crystals on thin films ofphotosensitive polymers—Effects of photoreactive group and UV-exposure,”Synth. Met., Vol. 117(1-3), pp. 273-5 (2001) (with these materials, theLC aligns nearly perpendicularly to the direction of polarization).Additional examples of methods of liquid crystal alignment are alsodiscussed in and U.S. Pat. No. 7,196,758 to Crawford et al. Furthermore,some structures described herein may involve precise fabrication througha balance of spin-coating processes and liquid crystal materials.Additional structures and/or methods for use with some embodiments ofthe present invention are discussed in PCT Publication No. WO2006/092758 to Escuti, et al., the disclosure of which is incorporatedby reference herein in its entirety.

Some embodiments of the present invention provide methods and devicesthat can achieve achromatic (broadband), high contrast diffraction usinga polarization grating having a twisted molecular structure along athickness thereof. For example, in some embodiments, high contrastachromatic diffraction may be achieved by using two liquid crystalpolarization grating layers of opposite twist sense that are laterallyoffset or shifted in phase relative to one another over their respectivethicknesses. More particularly, a first polarization grating with ahalf-wave retardation thickness and a +70° twist is laminated with asecond PG with a −70° twist on a transmissive substrate. In addition,when the first layer is embodied as a non-reactive liquid crystal layerand the second layer is embodied as a polymerizable liquid crystal layer(with respective twist angles of about +70° and about −70°), the gratingmay be switchable, and may provide a spatial-light-modulator suitablefor use in a liquid crystal display (LCD). In other embodiments, asingle polymer or non-reactive liquid crystal polarization grating layerhaving a 70° twist over a thickness thereof may be formed on areflective substrate to provide similar results. Other twist angles mayalso be used in any of the above embodiments. In contrast, while asingle-layer polarization grating may modulate unpolarized light, itshigh contrast operation may be limited to very narrow input light.Accordingly, as broadband light is present in many applications, someembodiments of the present invention may be used to providesubstantially higher contrast and/or brightness as compared to existingtechnologies.

In addition, in low-twist liquid crystal polarization gratings accordingto some embodiments of the present invention, it may be possible tobalance the chiral twist of the LC with the offset angle of thesubstrates in order to achieve an enhanced non-linear response of theelectro-optical curve. Accordingly, a less-expensive and/or lower-powerpassive matrix addressing scheme may be used, instead of an activematrix addressing scheme which may require a TFT within every pixel.Such an addressing scheme may offer significant advantages, for example,in portable applications.

FIGS. 1A to 1D illustrate polarization gratings according to someembodiments of the present invention. As shown in FIG. 1A, a secondpolarization grating layer PG2 102 is formed on a first polarizationgrating layer PG1 101 to form a multi-layer structure 105. The first andsecond polarization grating layers PG1 101 and PG2 102 are chiral liquidcrystal layers with molecular structures having an opposite twist senserelative to one another. In other words, the first and secondpolarization grating layers PG1 101 and PG2 102 include chiral molecules(i.e., asymmetric molecules having different left-handed andright-handed forms) of opposite handedness. As such, in someembodiments, the second polarization grating layer PG2 102 may have aphase shift of its local anisotropy pattern over a thickness d₂ oppositeto that of the first polarization grating layer PG1 101 over a thicknessd₁. The thicknesses d₁ and d₂ are respectively defined between opposingfaces of the first and second polarization grating layers PG1 101 andPG2 102.

More particularly, as shown in FIGS. 1B and 1C, the molecules of thesecond polarization grating layer PG2 102 are of an opposite handedness(left handed) as compared to the molecules of the first polarizationgrating layer PG1 101 (right handed). For example, the firstpolarization grating layer PG1 101 may be doped with a chiral moleculesuch that the orientation of the molecules therein may be rotated or“twisted” by a twist angle over the thickness d₁ of the layer PG1 101,and the second polarization grating layer PG2 102 may be doped withanother chiral molecule such that the orientation of the moleculestherein may be “twisted” by an opposite twist angle −θ_(twist) over thethickness d₂ of the layer PG2 102. In some embodiments, the secondpolarization grating layer PG2 102 may have a twist angle θ_(twist) ofabout −70°, while the first polarization grating layer PG1 101 may havea twist angle θ_(twist) of about 70°. In such embodiments, the thicknessd₁ of the first polarization grating layer PG1 101 may be substantiallyequal to the thickness d₂ of the second polarization grating layer PG1102. In fabricating the first and second polarization grating layers PG1101 and PG2 102, a nematic LC mixture may be doped with chiral LCmolecules configured to induce the respective twist angles thereinwithout substantial defects. The twist angle may be altered by varyingan amount of chiral dopant and/or a thickness of a polarization gratinglayer. The “twisting” of the molecules in each polarization grating overits thickness may provide a continuous phase-shifting in the localanisotropy pattern. As further illustrated in FIG. 1C, the molecules ofthe first and second polarization grating layers PG1 101 and PG2 102 arealigned or in-phase at the interface therebetween.

In some embodiments, the first and second polarization grating layersPG1 101 and PG2 102 may be single-substrate polymer layers, such asreactive mesogen (i.e., polymerizable liquid crystal) layers. Forexample, the first polarization grating layer PG1 101 may be formed byspin-casting a first chiral LC material (doped to provide apredetermined handedness or twist sense) on an exposed photo-alignmentlayer 115 such that it reaches the half-wave thickness for light used inoperation of the polarization grating. The photo-alignment layer 115 maybe formed and patterned on a transparent substrate, such as a glasssubstrate 110 a, by well-known techniques that will not be discussedfurther herein. A second chiral LC mixture doped to provide the oppositehandedness/twist sense may be directly applied on the first layer PG1101 until it also has the half-wave thickness to form the secondpolarization grating layer PG2 102.

In other embodiments, a switchable liquid crystal polarization gratingmay be formed. More particularly, a polarization grating layer PG2 102may be formed as described above with a predetermined handedness ortwist sense (for example, −70°). An opposing transmissive substrate(such as a glass substrate 110 a) including an exposed photo-alignmentmaterial 115 thereon may be laminated to the polarization grating layerPG2 102 with a cell gap corresponding to the half-wave cell thickness.The photo-alignment material 115 may include a periodic alignmentcondition that is offset based on the twist sense of the polarizationgrating layer PG2 102. The gap may be filled with a chiral nematic LCmaterial having the opposite twist sense (for example, +70°) to providea liquid crystal layer as the polarization layer PG1 101 between thephoto-alignment layer 115 and the polarization grating layer PG2 102 andthereby define the switchable liquid crystal polarization grating.

FIG. 1D illustrates a polarization grating according to furtherembodiments of the present invention. The polarization grating of FIG.1D includes a single polarization grating layer PG1 101 formed on asubstrate, such as a reflective substrate 110 b. For example, thepolarization grating layer PG1 101 may be formed on an exposedphoto-alignment layer 115 on the reflective substrate 110 b such that itreaches the half-wave thickness for light used in operation of thepolarization grating. As discussed above, the polarization grating layerPG1 101 may be doped with a chiral molecule such that the orientation ofthe molecules therein may be rotated or “twisted” by a twist angle overthe thickness d₁ of the layer PG1 101 to provide a continuousphase-shifting in the local anisotropy pattern. The polarization gratinglayer PG1 101 may have a twist angle of about 70°. However, the twistangle θ_(twist) may be altered by varying an amount of chiral dopantand/or a thickness of the polarization grating layer PG1 101. In someembodiments, the polarization grating layer PG1 101 may be apolymerizable liquid crystal layer, while in other embodiments, thepolarization grating layer PG1 101 may be a non-reactive liquid crystallayer to provide a switchable liquid crystal polarization grating.Because the operational light may pass through the polarization gratinglayer PG1 101 twice (upon both incidence and reflection) due to thepresence of the reflective substrate 110 b, the single-layerpolarization grating of FIG. 11) may optically function in a mannersimilar to that of the two-layer polarization grating of FIG. 1C, whichis further discussed in detail below.

Accordingly, some embodiments of the present invention providediffractive optical elements wherein the direction and/or polarizationstate of transmitted light may be controlled over a broad spectralrange. These diffractive optical elements may be used in displayapplications, for example, to provide more efficient outcoupling frombacklights, polarization-independent pixel designs, and/or lightrecycling.

Polarization gratings (PG) according to some embodiments of the presentinvention may be anisotropic periodic structures, can manifest uniquediffraction properties (such as three possible orders (0 and ±1) withspecial polarizations and up to 100% efficiency), and may support a widerange of applications. Conventional PGs may diffract with relativelyhigh efficiency over a spectral range of about 7% of a centerwavelength. In contrast, achromatic PGs according to some embodiments ofthe present invention may provide up to about a five-fold increase inthis bandwidth, and may diffract with up to about 100% efficiency over amajority of the visible spectral range, even with broadband illumination(e.g., white light). In particular, PGs according to some embodiments ofthe present invention may include at least two chiral liquid crystallayers, each having a relatively modest twist angle (such as 70°) andopposite twist sense.

Since the introduction of PGs as elemental polarization holograms, theirdiffraction properties and utility have been studied. For example,applications of PGs may include polarization measurement andhyperspectral polarimetry. Nematic liquid crystals (LCs) may createcontinuous-texture PGs with linear birefringence. Using this approach,substantially defect-free, switchable PGs may be created havingdesirable diffraction properties and/or relatively low scattering.Accordingly, switchable PGs may be used as polarization-independentmodulators.

A conventional (“circular”-type) PG may include a spatially-variantuniaxial birefringence (i.e., n(x)=[ cos(πx/Λ), sin(πx/Λ), 0]). Theideal diffraction efficiency at normal incidence can be derived asfollows:

$\begin{matrix}{{\eta_{0} = {{\cos^{2}\left( \frac{{\pi\Delta}\; {nd}}{\lambda} \right)}\mspace{14mu} {and}}}{\eta_{\pm 1} = {{\frac{1}{2}\left\lbrack {1 \mp S_{3}^{\prime}} \right\rbrack}{\sin^{2}\left( \frac{{\pi\Delta}\; {nd}}{\lambda} \right)}}}} & (1)\end{matrix}$

where η_(m) is the diffraction efficiency of the m^(th)-order, λ is thevacuum wavelength of incident light, Δn is the linear birefringence, dis the grating thickness, and S′₃=S₃/S₀ is the normalized Strokesparameter corresponding to ellipticity of the incident light. Threeorders (0 and ±1) may be present, and the first orders may possessorthogonal circular polarizations (left- and right-hand). Thediffraction behavior of PGs may depend on the wavelength (through Δnd/λin Eq. (1)).

Referring again to FIGS. 1B and 1C, the achromatic performance of a PGincluding a two-layer twisted structure according to some embodiments ofthe present invention may provide up to 100% efficiency across arelatively wide spectral width, for example, up to about 34.3% of acenter wavelength. This represents an increase by about a factor of fiveas compared to that of conventional PGs, which may provide a spectralwidth of about 6.8% of the center wavelength, Accordingly, achromacityof PG diffraction can be achieved by combining two twisted PGs withopposite twist sense,

Some design parameters for broadband diffraction of twisted PGs mayinclude the thickness d and the twist angle θ_(twist) of each PG layer.Effects of these parameters have been demonstrated using thefinite-difference time-domain (FDTD) method and an open-source softwarepackage especially developed for periodic anisotropic media.Accordingly, preliminary experimental results are discussed below withreference to FIGS. 2-6 for achromatic PGs formed as a polymerizableliquid crystal film using polarization holography and photo-alignmenttechniques.

FIG. 2 illustrates the basic geometry of the FDTD simulation spacedescribed above. Gradient-index anti-reflection (AR) coatings 206 may beapplied to the polarization grating PG 201 at both air-polarizationgrating interfaces to reduce and/or minimize Fresnel losses. Periodicboundaries 207 and matched layer boundaries 208 using the UniaxialPerfectly Matched Layer (UPML) technique may be employed to terminatethe simulation space and/or reduce simulation time. The input/incidentplanewave 209 may be a Gaussian-pulsed planewave (i.e., a widebandsource) with vertical-linear polarization placed just before the gratingstructure, and the output diffraction efficiencies may be calculatedfrom the electric field at a line 211 immediately after the grating. Anear-to-far optical transformation and a temporal Fourier transform maybe used to analyze spectral diffraction properties in the far-field.

As used herein, the spectral range Δλ (in units of wavelength) for highPG efficiency is defined as the range of wavelengths over which thetotal first-order diffraction Ση_(±1) is greater than about 99.5%. Thenormalized bandwidth Δλ/λ_(center) (in units of %) is defined as theratio of the spectral range to its center wavelength λ_(center).

FIG. 3 illustrates simulation results showing diffraction properties ofa single-layer twisted PG for a range of different twist angles from 0°to 90°. As such, the data may be the same for right- andleft-handedness. More particularly, FIG. 3 illustrates a sum of thefirst-order efficiency (Ση_(±1)) versus normalized retardation (Δnd/λ)for different twist angles (θ_(twist))0°, 30°, 60°, 70°, and 90°,respectively represented by waveforms 301, 302, 303, 304, and 305. Asshown in FIG. 3, a maximum high-efficiency bandwidth occurs in the caseof a conventional PG (i.e., illustrated by waveform 301 whereθ_(twist)=0°, and results in Δλ/λ_(center)=6.8%. Accordingly, since thecondition for adiabatic-following (also known as waveguiding) may not bemet except for very small twist angles, a degradation in efficiency mayresult with increasing twist angles. Nevertheless, only the 0 and±1-orders are present in the output and the first-order polarizationsbecome increasingly elliptical (as opposed to circular).

However, high diffraction efficiency (i.e., up to about 100%) can beprovided according to some embodiments of the present invention bystacking two twisted PGs with opposite twist sense. The opticalproperties of both PG layers may be substantially similar or identical(except for the direction of twist). As such, the second layer maycompensate for the polarization effect of the first twisted structure.Therefore the achromatic effect may be qualitatively described aslocalized retardation compensation.

FIG. 4A illustrates simulation results showing diffraction properties ofa two-layer twisted structure with opposite twist sense according tosome embodiments of the present invention with varying twist angles overa range of about 0° to about 90°. More particularly, FIG. 4A shows thefirst order efficiency (Ση_(±1)) as a function of normalized retardation(Δnd/λ) for different values of 0°, 30°, 60°, 70°, and 90°, respectivelyrepresented by waveforms 401, 402, 403, 404, and 405. The maximumbandwidth Δλ/λ_(max)=34.3% can be achieved when θ_(twist)=70°, asfurther illustrated in FIG. 4B. More particularly, the gray scale levelsof FIG. 4B illustrate simulated diffraction efficiency, and thebandwidth Δλ/λ_(center) is at a maximum over the region shown.Accordingly, about a five-fold enhancement in the maximum diffractionbandwidth may be achieved as compared with a conventional PG. Since thediffraction bandwidth may be sensitive to the twist angle, carefulcontrol of θ_(twist) may be important to provide improved bandwidthperformance.

In some embodiments, achromatic PGs according to some embodiments of thepresent invention may be formed as a polymerizable liquid crystal filmusing a combination of polarization holography and photo-alignmenttechniques. Substantially defect-free RM PGs may be fabricated withrelatively high efficiency and/or low scattering based on materials andprocessing optimization. As illustrated in FIGS. 5A-5E, fabrication ofpolymerizable liquid crystal PGs 501 and 502 may proceed as follows. Asshown in FIG. 5A, a relatively thin layer of photo-alignment material515 is coated on a substrate 505. The substrate 505 may be atransmissive or transparent substrate, such as a glass substrate, insome embodiments. However, in other embodiments, the substrate 505 maybe a reflective substrate. The substrate 505 is exposed or patternedusing coherent beams 509 from a laser with orthogonal circularpolarizations at a relatively small angle to provide a polarizationinterference pattern 516 with a substantially constant intensity, asshown in FIG. 5B. In FIG. 5C, a first RM layer 501 having a first twistsense is formed on the photo-alignment layer 515 and is alignedaccording to the surface pattern 516. For example, a first RM mixturemay be doped with a first chiral dopant and spin-cast onto thephoto-alignment layer to provide the first RM layer 501. The first RMlayer 501 is photo-polymerized, for example, using a blanket ultraviolet(UV) exposure 519, to permanently fix the large structured opticalanisotropy, as shown in FIG. 5D. In FIG. 5E, a second RM layer 502having an opposite twist sense is formed on the first RM layer 501. Forexample, a second RM mixture may be doped with a second chiral dopant,spin-cast onto the first RM layer 501, and photo-polymerized to providethe second RM layer 502. The second RM layer 502 is aligned based on thealignment of the first RM layer 501 at the interface therebetween.

Still referring to FIGS. 5A-5E, in some embodiments, alinear-photopolymerizable polymer (LPP), such as ROP-103 (Rolic), may beused as the photo-alignment material 515. A HeCd laser (325 nm) withorthogonal circular polarized beams may be used to expose or form asurface alignment pattern with a period of Λ=8.5 μm onto thephoto-alignment layer 515. After photo-alignment exposure, the first andsecond RM films 501 and 502 may be deposited on the photo-alignmentlayer 515 on the substrate 505 by spin-coating. The first RM layer 501may be a mixture composed of RMS03-001 (Merck, Δn˜0.159 at 589 nm) witha small amount (0.25%) of chiral dopant CB15 (Merck, right-handedness),and may be chosen so that the thickness d₁ of the first RM layer 501reaches the half-wave thickness (d=λ/2Δn) and θ_(twist)=70°. The secondRM layer 502 may be deposited directly on top of the first RM layer 501,and may be composed of RMS03-001 doped with a small amount (0.34%) of adifferent chiral dopant ZLI-811 (Merck, left-handedness) subject to thesame thickness and an opposite twist condition. Accordingly, the finalgrating thickness of a polarization grating according to someembodiments of the present invention may be 2d, because the two layers501 and 502 may be stacked, each having a thickness of about thehalf-wave thickness d.

FIG. 6A provides experimental results illustrating 0-order efficiencyspectra for a conventional PG (shown by waveform 610) and an achromaticPG according to some embodiments of the present invention (shown bywaveform 620) as measured with a spectrophotometer. The measuredtransmittance of a clean glass slide (i.e., about 100%; shown bywaveform 630) is also provided, and was measured under substantiallysimilar conditions to the PGs. The spectra of the estimated diffractionefficiency calculated from the 0-order (Ση_(±1)≈1−η₀) for a conventionalPG (610′) and for an achromatic PG (620′) according to some embodimentsof the present invention is plotted in FIG. 6B. As expected from theFDTD simulation results of FIGS. 3 and 4A-4B, a noticeable improvementin the diffraction bandwidth is illustrated in FIGS. 6A and 6B.Efficiencies at three wavelengths were also measured with red (633 nm),green (532 nm), and blue (473 nm) lasers for both the conventional PGand the achromatic PG according to some embodiments of the presentinvention to confirm the estimated efficiencies shown in FIG. 6B. Asused herein, the diffraction efficiency is defined asη_(m)=I_(m)/I_(REF), where I_(m) is the measured intensity of the m^(th)transmitted diffracted order, and where I_(REF) is a referencetransmission intensity for a glass substrate. The incoherent scatteringwas roughly measured as about 2% or less above 400 nm by comparing thediffracted spectra to the clean glass slide.

Accordingly, achromatic PGs including at least two layers with oppositetwist sense according to some embodiments of the present invention mayprovide diffraction properties such as three diffracted orders (0, ±1),orthogonal circular polarizations of the first orders, and/or highlypolarization-sensitive first orders (which may be linearly proportionalto the Stokes parameter). In addition, incident circular polarizationcan produce up to about 100% efficiency into one of the first orders,and linear incident polarization or unpolarized input can give up toabout 50% efficiency into each of the first orders.

Compared to other LC gratings (i.e. polymer-wall LC gratings and/orHPDLC gratings), achromatic PGs according to some embodiments of thepresent invention may provide comparable or higher experimentaldiffraction efficiencies, and/or lower incoherent scattering. As such,achromatic PGs according to some embodiments of the present inventionmay offer the high efficiencies of thick (Bragg) gratings over nearlythe entire range of visible light. When used as optical elements indisplays, achromatic PGs according to some embodiments of the presentinvention may be integrated with other optical components, which mayresult in more compact and efficient displays. These diffractive opticalelements may also be useful for beamsplitting, polarimetry, and more. Inaddition, a similar achromatic PG design can be implemented to provide aswitchable LC grating for modulator applications. More particularly, oneof the two twisted PG layers may be implemented with a non-reactivenematic liquid crystal material, and the entire structure may be placedbetween substrates with electrodes to provide the switchable LC grating.

Achromatic PGs including at least two layers with opposite twist senseaccording to some embodiments of the present invention may therebyachieve relatively high efficiency over a broad spectral range, and assuch, may offer a wide range of potential applications in displaytechnologies to provide more efficient control of light based its uniquediffraction behavior. More particularly, such thin-film achromatic PGsmay offer substantially more functional control than conventionaldiffraction gratings over the direction, intensity, and/or polarizationstate of the transmitted light (for a wide spectral bandwidth), and mayoffer potential benefits in many remote sensing applications.

In addition, achromatic PGs according to some embodiments of the presentinvention can be fabricated using known thin-film techniques and/orknown liquid crystal materials to create an improvedspatial-light-modulator element, for example, for use inpolarization-independent microdisplays such as portable projectiondisplays, consumer TV sets, real-time holography, etc. Moreover, in someinstances, achromatic PGs according to some embodiments of the presentinvention may increase high contrast modulation over a wavelength rangeof up to about 45% of a desired center wavelength, which may result inup to a 900% improvement as compared to conventional single polarizationgratings. Accordingly, in display applications, image quality may besignificantly improved.

Furthermore, achromatic PGs according to some embodiments of the presentinvention may be used to provide an imaging polarization interferometer.As the PG may produce three diffracted orders and each of the firstorders may have polarization sensitivity, a combination of the PG and awaveplate may allow separation of orthogonal polarization informationinto the two diffracted orders. Accordingly, a compactspectropolarimeter may be provided by combining this polarizationselectivity and the chromatic dispersion of the PG. For example, insteadof measuring intensities of each diffracted orders, the samepolarization information (i.e. the Stokes parameters) can be extractedfrom the interferogram of two diffracted beams traveling through twosubstantially identical PGs. More particularly, the first PG mayseparate beams containing spectral images into two first orders, and thesecond PG may re-direct beams in parallel. The polarization state ofeach diffracted beam may be converted to the same linear polarizationafter traveling λ/2-waveplates oriented at a right angle with eachother. The interference pattern can be obtained by focusing twodiffracted beams on the same image plane where a detector locates.

Further embodiments of the present invention provide switchable liquidcrystal polarization gratings (LCPGs) that may be controlled at thepixel-level using a passive matrix addressing scheme. Passive matrixaddressing may employ a row-and-column voltage-averaging approach toreduce and/or eliminate the need for a thin-film-transistor (TFT) withineach individual pixel, as used in higher-cost active matrix addressing.Accordingly, passive matrix addressing may be used in LCDs wherelow-power and low-cost are required (such as cell phones and PDAs).However, implementation of passive matrix addressing in LCPGs mayrequire a relatively steep electro-optic response curve. In contrast,conventional LCPG technology may have a relatively poor electro-opticresponse curve, and as such, may not be suitable for use with passivematrix addressing. Accordingly, some embodiments of the presentinvention provide switchable LCPGs with significantly steeper (in slope)electro-optical response curves, which may improve the number of rowsthat can be passively addressed from about 1 row to more than about 100rows. This may be comparable to conventional cell-phone LCDs based onthe super-twisted-nematic (STN) configuration.

FIGS. 7A-7D are cross-sectional views illustrating methods offabricating LCPGs and devices so fabricated according to furtherembodiments of the present invention. Referring now to FIG. 7A,relatively thin alignment layers 715 a and 715 b are respectively formedon first and second substrates 705 and 710. The first and/or secondsubstrates 705 and/or 710 may be formed of a transmissive or transparentmaterial, such as glass. Each substrate may also include transparentconductive electrodes (not shown). In FIG. 7B, the alignment layers 715a and 715 b on each substrate are patterned so as to provide periodicalignment conditions on each substrate. For example, the alignmentlayers may be photo-alignment layers including photopolymerizablepolymers therein, and may be holographically patterned using orthogonalcircularly polarized laser beams 709 a and 709 b. The first and secondsubstrates 705 a and 710 b are assembled as shown in FIG. 7C such thatthe periodic alignment conditions of the respective alignment layers 715a and 715 b are offset by a relative phase angle Φt. Accordingly, asshown in FIG. 7D, a liquid crystal layer 725 having a predeterminedtwist sense is formed in the cell gap 721 between the first and secondsubstrates. The liquid crystal layer may be a single layer of nematic LCdoped with a chiral molecule to provide a particular twist Φc over thethickness d of the cell-gap 721. In other words, the twist angle Φc mayprovide a continuously variable phase-shift in the local anisotropypattern over the thickness d of the liquid crystal layer. The moleculesof the liquid crystal layer 725 may also be aligned based on thealignment conditions in the photo-alignment layers. Accordingly, whenthe twist angle Φc is different from the substrate offset angle Φt, anelastic-energy strain can be generated in the liquid crystal layer,which may result in a more non-linear switching behavior. As with STNdisplays, this may be quantified by calculating the mid-layer tilt angleas a function of applied voltage, as further illustrated in the examplesimulation results of FIGS. 8A-8E.

FIGS. 8A-8E illustrate the electro-optic response of LCPGs according tosome embodiments of the present invention. As shown in FIGS. 8A-8E, asthe twist angle Φc is changed relative to the substrate offset angle Φt,a relatively steep electro-optic response may be obtained. Moreparticularly, as illustrated in FIG. 8A, the baseline curves (Φt=0°,Φc=0° to 89°) may not be suitable for use with passive addressing, sinceit may enable only about 1 row with relatively good contrast. As furtherillustrated in FIG. 8B, in the case of the (Φt=300°, Φc=240° to 345°)design, the curves are significantly more steep. Accordingly, more than100 rows may be passively-addressed. FIGS. 8C and 8D similarlyillustrate the electro-optic response for varying values of Φc whenΦt=240° and 270°, respectively. Additional details regarding the aboveestimates may be found in the summary of STN display addressing inSchafer and Nehring, Annual Review of Material Science 27, 555-583(1997) and in Alt and Pleshko, IEEE Trans. Elec, Dev. ED-21, 146-155(1974), the disclosures of which are incorporated by reference herein.

FIG. 8E illustrates the electro-optic response curves of FIGS. 8A, 8B,and 8C along a common set of axes for comparison. In particular, curve805 a illustrates the electro-optic response for the case where Φt=0°and Φ_(c)=0°. Likewise, curve 805 b shows the electro-optic response forthe case where Φt=300° and Φc=240°, while curve 805 c illustrates thecase where Φt=240° and Φc=180°. Accordingly, FIG. 8E illustrates thatthe response curves can be made significantly more steep based onchanges in the twist angle (be and the substrate offset angle Φt. Inparticular, the response curve 805 b provides a relatively steep slopefor improved switching behavior where the relative phase angle Φt isgreater than or equal to about 300°, and where the twist angle Φc isbetween about 240° and about 300°. More generally, in some embodiments,the relative phase angle Φt may be about 70° to about 360°, and thetwist angle Φc may be about 70° to about 360°.

Accordingly, LCPGs according to some embodiments of the presentinvention may use a chiral strain and a twist structure to provide amore non-linear electro-optical response curve. Thus, LCPGs according tosome embodiments of the present invention may more readily switch fromoff- to on-states, and as such, may be more controllable using passivematrix addressing schemes.

The foregoing is illustrative of the present invention and is not to beconstrued as limiting thereof. Although a few exemplary embodiments ofthis invention have been described, those skilled in the art willreadily appreciate that many modifications are possible in the exemplaryembodiments without materially departing from the novel teachings andadvantages of this invention. For example, it is to be understood thatthe structures described above with reference to FIGS. 1A-1D and FIGS.7A-7D can be fabricated using non-switchable and/or switchable LCmaterials, in one or two substrate assemblies, respectively. Moreover,the substrates described herein may include one or more electrodes onsurfaces thereof, for instance, provided by a transparentindium-tin-oxide (ITO) coating on the substrates. Accordingly, all suchmodifications are intended to be included within the scope of thisinvention. Therefore, it is to be understood that the foregoing isillustrative of the present invention and is not to be construed aslimited to the specific embodiments disclosed, and that modifications tothe disclosed embodiments, as well as other embodiments, are intended tobe included within the scope of the invention.

What which is claimed:
 1. An optical element, comprising: first andsecond stacked birefringent layers having respective local optical axesthat are rotated by respective twist angles over respective thicknessesdefined between opposing faces of the first and second birefringentlayers, wherein the respective twist angles are different.
 2. Theoptical element of claim 1, wherein the respective twist angles areopposite in twist sense.
 3. The optical element of claim 2, wherein therespective twist angles are substantially equal in magnitude.
 4. Theoptical element of claim 3, wherein the respective thicknesses aresubstantially equal.
 5. The optical element of claim 1, wherein therespective local optical axes of the first and second layers are definedby local anisotropy patterns having respective orientations that varyover the respective thicknesses of the first and second layers.
 6. Theoptical element of claim 5, wherein the respective orientations of thelocal anisotropy patterns of the first layer continuously vary relativeto one another in a manner opposite to those of the local anisotropypatterns of the second layer over the respective thicknesses of thefirst and second layers.
 7. The optical element of claim 1, wherein therespective local optical axes of the first and second layers are definedby molecules having respective relative orientations that are rotatedover the respective thicknesses of the first and second layers to definethe respective twist angles.
 8. The optical element of claim 7, whereinthe molecules of the first and second layers comprise nematic liquidcrystal molecules.
 9. The optical element of claim 8, wherein themolecules of the first and second layers comprise chiral molecules ofopposite handedness.
 10. The optical element of claim 7, wherein atleast one of the first and second layers comprises a polymerized liquidcrystal layer.
 11. The optical element of claim 10, wherein another ofthe first and second layers comprises a non-reactive liquid crystallayer.
 12. The optical element of claim 7, wherein the respectiverelative orientations of the molecules of the first and second layersare rotated by the respective twist angles throughout the first andsecond layers.
 13. The optical element of claim 1, wherein therespective local optical axes of the first and second layers are alignedalong an interface therebetween.
 14. The optical element of claim 13,wherein the respective local optical axes of the first and second layersvary in a direction along the interface therebetween.
 15. The opticalelement of claim 14, wherein the first and second layers comprisepolarization gratings including local anisotropy patterns havingrespective orientations that vary periodically in a direction along theinterface therebetween.
 16. The optical element of claim 1, wherein thefirst and second layers are stacked directly on one another and define amonolithic structure.
 17. The optical element of claim 1, furthercomprising: an alignment surface having a varying alignment conditiontherein, wherein the respective local optical axes of the first layerare oriented according to the alignment condition along an interfacebetween the alignment surface and the first layer.
 18. The opticalelement of claim 17, wherein the alignment condition comprises aperiodic alignment condition.
 19. The optical element of claim 1,wherein the respective thicknesses of the first and/or second layers areconfigured to provide half-wave retardation of light within anoperational wavelength range of the optical element.
 20. An opticalelement, comprising: a first birefringent layer comprising localanisotropy patterns having respective relative orientations that varyover a first thickness between opposing faces of the first birefringentlayer to define a first twist angle; and a second birefringent layer onthe first birefringent layer, the second birefringent layer comprisinglocal anisotropy patterns having respective relative orientations thatvary over a second thickness between opposing faces of the secondbirefringent layer to define a second twist angle different than thefirst twist angle.
 21. The optical element of claim 20, wherein thefirst and second twist angles are opposite in sign.
 22. The opticalelement of claim 21, wherein the first and second twist angles aresubstantially equal in magnitude.
 23. The optical element of claim 20,wherein the respective relative orientations of the local anisotropypatterns of the first and second birefringent layers are aligned alongan interface therebetween.
 24. The optical element of claim 23, whereinthe respective relative orientations of the local anisotropy patterns ofthe first and second birefringent layers vary in a direction along theinterface therebetween.
 25. A method of fabricating an optical element,the method comprising: providing a first birefringent layer; andproviding a second birefringent layer on the first birefringent layer,wherein the first and second birefringent layers have first and secondlocal optical axes that are rotated by first and second twist anglesover first and second thicknesses thereof, respectively, and wherein thefirst and second twist angles are different.
 26. The method of claim 25,wherein the first and second twist angles are opposite in sign.
 27. Themethod of claim 26, wherein the first and second twist angles aresubstantially equal in magnitude.
 28. The method of claim 27, whereinthe first and second thicknesses are substantially equal.
 29. The methodof claim 25, wherein providing the first and second layers comprises:forming the first layer comprising local anisotropy patterns havingrespective relative orientations that vary over the first thickness todefine the first twist angle; and forming a second layer on the firstlayer, the second layer comprising local anisotropy patterns havingrespective relative orientations that vary over the second thickness todefine the second twist angle, wherein the local anisotropy patterns ofthe first and second layers define the first and second local opticalaxes thereof, respectively, and wherein the respective orientations ofthe local anisotropy patterns of the first and second layers are alignedalong an interface therebetween.
 30. The method of claim 29, wherein thefirst and second layers are liquid crystal layers comprising nematicliquid crystal molecules that define the local anisotropy patternsthereof, and wherein forming the second layer comprises: forming thesecond layer directly on the first layer such that the respectiverelative orientations of the nematic liquid crystal molecules of thesecond layer are aligned according to the respective relativeorientations of the nematic liquid crystal molecules of the first layeralong the interface therebetween.
 31. The method of claim 30, wherein atleast one of the first and second layers comprises a polymerized liquidcrystal layer.
 32. The method of claim 31, wherein another of the firstand second layers comprises a non-reactive liquid crystal layer.
 33. Themethod of claim 29, wherein the respective orientations of the localanisotropy patterns of the first and second layers vary in a directionalong the interface therebetween.
 34. The method of claim 33, whereinthe first and second layers comprise polarization gratings, and whereinthe respective orientations of the local anisotropy patterns of thefirst and second layers vary periodically in the direction along theinterface therebetween.
 35. The method of claim 29, wherein forming thefirst and second layers comprises: doping the first liquid crystal layerwith a chiral molecule having a first handedness; and doping the secondlayer with a chiral molecule having a second handedness opposite to thefirst handedness.
 36. The method of claim 29, wherein the respectiverelative orientations of the local anisotropy patterns of the first andsecond layers are rotated by the first and second twist anglesthroughout the first and second layers, respectively.
 37. The method ofclaim 25, further comprising: forming an alignment surface having avarying alignment condition therein; and forming the first layerdirectly on the alignment surface such that the respective local opticalaxes of the first layer are oriented according to the alignmentcondition along an interface between the alignment surface and the firstlayer.
 38. The method of claim 37, wherein the alignment conditioncomprises a periodic alignment condition, and wherein providing thesecond layer further comprises: forming the second layer directly on thefirst layer such the respective local optical axes thereof are alignedaccording to the respective local optical axes of the first layer alongan interface therebetween.
 39. The method of claim 37, wherein the firstlayer comprises a polymerizable liquid crystal layer, and furthercomprising: photo-polymerizing the first layer on the alignment surfaceprior to providing the second layer on the first layer.